Objective 1 - Holes

$\newenvironment {prompt}{}{} \newcommand {\ungraded }[0]{} \newcommand {\npnoround }[0]{\nprounddigits {-1}} \newcommand {\npnoroundexp }[0]{\nproundexpdigits {-1}} \newcommand {\npunitcommand }[1]{\ensuremath {\mathrm {#1}}} \newcommand {\HyperFirstAtBeginDocument }[0]{\AtBeginDocument }$

Link to section in online textbook.

Introduction video describing holes/vertical asymptotes without limits.

A hole in a function occurs when the value of that function is $\frac {0}{0}$ . For example, the function

$f(x) = \frac {(x+2)(x-3)}{x-3}$

has a hole at $x=3$ because $f(3) = \frac {0}{0}$ . If we want to describe this with limits, we would say $\lim _{x \rightarrow 3} f(x) = \frac {0}{0}$ . Holes only affect the function exactly at that point. Notice for our example that

$f(x) = x+2$ when $x \neq 3$ and $f(x)$ is undefined at $x=3$ .

That means the rational function actually looks like a line almost everywhere! Recognizing if a rational function has holes will be our first step in graphing these functions.

Holes of a Rational Function:

A rational function $f(x)$ has a hole at $x=a$ if

$\lim _{x \rightarrow a} f(x) = \frac {0}{0}.$

Thus, to determine if there are any holes in a rational function, we need to factor the denominator and check if that value is a zero of the numerator (using Synthetic Division, if necessary).

Practice with the questions below.